The
situation is much more complicated if the original theory has the
pure Weyl-squared term in the Lagrangian, i.e., possesses local
conformal symmetry. The theoretical proof of the renormalizability
in conformal theories exists only for the quantum theory in
curved space (semiclassical gravity) [1]} and, in the
literature, there is no proof for conformal QG. On the other hand,
at the one-loop order this theory is renormalizable, as was shown
by direct calculations [2]}, [3]} and confirmed in
[4]}.
However, it is expected that this model is not renormalizable at
higher loops because of the conformal anomaly. But, in the
situation when this expectation is not supported by direct
higher-loop calculations or the analysis similar to [1]},
the question should be regarded as open.
